Two-sided sgut-majorization and its linear preservers

Let $\textbf{M}_{n,m}$ be the set of all $n$-by-$m$ real matrices, and let $\mathbb{R}^{n}$ be the set of all $n$-by-$1$ real vectors. An $n$-by-$m$ matrix $R=[r_{ij}]$ is called g-row substochastic if $\sum_{k=1}^{m} r_{ik}\leq 1$ for all $i\ (1\leq i \leq n)$. For $x$, $y \in \mathbb{R}^{n}$, it is said that $x$ is $\textit{sgut-majorized}$ by $y$, and we write $ x \prec_{sgut}y$ if there exists an $n$-by-$n$ upper triangular g-row substochastic matrix $R$ such that $x=Ry$. Define the relation $\sim_{sgut}$ as follows. $x\sim_{sgut}y$ if and only if $x$ is sgut-majorized by $y$ and $y$ is sgut-majorized by $x$. This paper characterizes all (strong) linear preservers of $\sim_{sgut}$ on $\mathbb{R}^{n}$.